Math 362: Real and Abstract Analysis 2
Spring 2013, Professor Levandosky
Solutions to Final Exam Practice Questions
1. (a) Show that if
f (x) dx = 0 for every measurable subset A of R, then f = 0
A
almost everywhere.
Solution. Taking A = R we get
f (x) dx
Math 362: Real and Abstract Analysis 2
Spring 2013, Professor Levandosky
Solutions to Exam 3 Practice Problems
1. Let
f (x) =
en x sin(nex ).
2
n=1
Show that f L ([0, ).
1
2
Let fn (x) = en x sin(nex ). It suces to show that the series
fn L1 converges.
Si
Math 362: Real and Abstract Analysis 2
Spring 2013, Professor Levandosky
Solutions to Exam 2 Practice Questions
1. Let A be the subset of (0, 1) consisting of all real numbers whose decimal expansion
does not contain the digit 5. Find m(A).
Solution. The
Math 362: Real and Abstract Analysis 2
Spring 2013, Professor Levandosky
Solutions to Exam 1 Practice Problems
1. Let T : R2 R2 be the linear transformation with matrix
[
]
12
A=
32
Find T .
Solution. The eigenvalues of
[
At A =
are = 9
65, so T = max =
Math 362: Real and Abstract Analysis
College of the Holy Cross, Spring 2013
Homework 7 Solutions
1. For which p are the following functions in Lp (R)?
(a) f = x11/7 (0,1) + x11/3 (1,)
Solution. Let f1 = x11/7 (0,1) and f2 = x11/3 (1,) . Then f Lp if and o
Math 362: Real and Abstract Analysis
College of the Holy Cross, Spring 2013
Homework 6 Solutions
1. Show that convolution is associative: f (g h) = (f g ) h.
Solution. First write
(f (g h)(x) =
f (x y )(g h)(y ) dy
=
f (x y ) g (y z )h(z ) dz dy
R
R
=
f (
Math 362: Real and Abstract Analysis
College of the Holy Cross, Spring 2013
Homework 5 Solutions
1. (a) Give an example of a set S and functions f, g L1 (S ) such that the product f g
is not in L1 (S ).
Solution. Let S = [0, 1] and let f (x) = g (x) = x1/
Math 362: Real and Abstract Analysis
College of the Holy Cross, Spring 2013
Homework 4 Solutions
1. Suppose m(A) = 0.
(a) Show that if is any measurable simple function then
dx = 0.
A
n
Solution. Let = k=1 ak Ak be simple and measurable. Then
dx =
A dx
Math 362: Real and Abstract Analysis
College of the Holy Cross, Spring 2013
Homework 3 Solutions
1. Suppose Z is a zero set.
(a) Show that Z is measurable.
We need to show that for any subset X of Rn , m (X ) = m (X Z ) + m (X Z c ).
By subadditivity,
m (
Math 362: Real and Abstract Analysis
College of the Holy Cross, Spring 2013
Homework 2 Solutions
1. (a) Show that the equation sin(x + y ) + xy = 0 denes a function y = g (x) implicitly
for (x, y ) in some neighborhood of (0, 0).
Solution. Let f (x, y ) =
Math 362: Real and Abstract Analysis
College of the Holy Cross, Spring 2013
Homework 1 Solutions
1
1
1. Let T : R2 R3 be the linear transformation with matrix A = 1 1. Compute
1 1
T , and nd a vector in v R2 such that |T (v )| = T |v |.
Solution.
]1
[
]
1
Math 362: Real and Abstract Analysis
College of the Holy Cross, Spring 2013
Exam 3 Solutions
Professor Levandosky
1. For which p is the function f (x, y ) =
For p = 1,
R2
in L1 (R2 )?
1
(1+x2 +y 2 )p
2
|f (x, y )| dA =
0
0
1
r dr d
(1 + r2 )p
1 (1 + r2 )p
Math 362: Real and Abstract Analysis
College of the Holy Cross, Spring 2013
Exam 2 Solutions
Professor Levandosky
1. Complete the following denitions.
(a) The Lebesgue outer measure of a subset A of Rn is
m (A) = inf
cfw_
|Ik | : A
k=1
Ik
.
k=1
(b) A su
Math 362: Real and Abstract Analysis
College of the Holy Cross, Spring 2013
Exam 1 Solutions
Professor Levandosky
1. Dene T : R2 R3 to be the linear transformation with matrix
21
A = 1 1 .
1 3
Compute T and nd a vector v such that |T (v)| = T |v|.
[
Solut